On the question: In what ways can sampling strategy impact the uncertainty that is associated with using samples to make inferences about populations? Perhaps an example may assist, where I believe the impact is somewhat intuitive.
Consider, for example, a case where one is interested in an estimate of spending by consumers on luxury goods. The most accurate sampling scheme could be a stratified sampling model with a particular sampling concentration on high-income individuals. Why? Because there is an inherent bias for more affluence people to engage in the purchase of luxury goods (likely somewhat proportional to their income). This would likely account for a large part of total spending on luxury goods, and thus would assist in estimating total sector spending. The latter is the targeted purpose of the sampling exercise and studying/expending effort (as could occur, for example, in a Simple Random Sampling of the general population) on the spending patterns of those who rarely buy luxury goods is not likely as useful in improving the precision of the sampling scheme's luxury spending estimate.
Note: Actually knowing the percentage of higher-income people in the general population, however, is fundamental to the appropriate application of stratified sampling. For the precise formula for stratified sampling, including its expected sampling variability, see, for example, this reference, and related comments.
Another sampling scheme, contextually removed from stratified sampling and worthy of mention, in my opinion, is Systematic Sampling. Some major points of the latter includes ease of implementation and understanding,
an associated degree of control and sense of underlying process, ability to address problematic, for example, clustered selection presence, and lastly, generally considered a low-risk process.
Note: Systematic Sampling generally assumes size of the parent population can be determined, and also requires some degree of naturally occurring randomness. However, it does afford a greater risk of data manipulation (as from examining varying results per the changing of sample composition by selecting from every nth item to some other count).